Elementary Singular Point Research Articles

Complete classification on center of cubic planar systems symmetric with respect to a straight line

In this paper, we study bi-center and bi-isochronous center problems in cubic planar systems which are symmetric with respect to a straight line. These systems can be transformed to ones which are symmetric with respect to the y-axis and have two symmetric singular points at (±1,0), which can be classified as elementary and nilpotent singular points. A complete classification is given on the centers, including nine conditions for elementary singular points and four conditions for nilpotent singular points. Moreover, seven bi-isochronous center conditions are obtained for the elementary singular points.

Limit Cycles and Integrability of a Class of Quintic System

In this paper, a class of quintic systems is investigated. The first 13 focal values are computed with the aid of MATHEMATICA. Then the necessary conditions of integrability and linearizability are obtained and the sufficiency of every condition is proved. Meanwhile, bifurcation of limit cycles is discussed, 13 limit cycles can be bifurcated from the origin. As far as the number of limit cycles enclosing an isolated singular point is concerned, this is so far the best result for elementary singular points.

Integrability and linearizability of cubic Z2 systems with non-resonant singular points

In this paper, complete integrability and linearizability of cubic Z2 systems with two non-resonant and elementary singular points are investigated. First of all, four simple normal forms are obtained based on the coefficients and eigenvalues of cubic Z2 systems. Then, the integrable and linearizable conditions are classified according to the four different cases respectively, and the problem is solved thoroughly for cubic Z2 systems with two non-resonant singular points.

A note on the paper “Center and isochronous center conditions for switching systems associated with elementary singular points”

In this note, we correct an error in our paper published in Communications in Nonlinear Science and Numerical Simulation in 2015 [5], in which the form of a Z2-equivariant cubic planar switching system is not correct. In this note, we add one more switching line and use the same equations given in [5] to construct a correct Z2-equivariant cubic system. Then, we prove that the new system can exhibit 15 limit cycles and therefore the conclusion given in [5] still holds.

Bifurcation analysis on a class of Z2-equivariant cubic switching systems showing eighteen limit cycles

In this paper, the method developed for computing the Lyapunov constants of planar switching systems associated with an elementary singular point is applied to study bifurcation of limit cycles in a cubic switching system. A complete classification on the center conditions and 16 limit cycles of this system are obtained around the two foci (1,0) and (−1,0). Further, with the method, an example of cubic switching systems is constructed to show the existence of 18 small-amplitude limit cycles bifurcating from centers. This is a new lower bound on the maximal number of small-amplitude limit cycles obtained in such cubic switching systems. Finally, a method is present to show the realization of the 18 limit cycles.

Введение в теорию нестабильности на примере особых точек начальной задачи процесса деформирования сжатых стержней в условии ползучести

Введение в теорию нестабильности на примере особых точек начальной задачи процесса деформирования сжатых стержней в условии ползучести

Center and isochronous center conditions for switching systems associated with elementary singular points

In this paper, an existing method is modified for computing the focal values and period constants of switching systems associated with elementary singular points. In particular, a quadratic switching system is considered to illustrate the computational efficiency of this method. Further, with this method, a cubic switching system is constructed to show existence of 15 limit cycles, which is the best result so far obtained for cubic switching systems.

Integrability and Pseudo‐Linearizable Conditions in a Quasi‐Analytic System

This paper deals with the problems of integrability and linearizable conditions at degenerate singular point in a class of quasianalytic septic polynomial differential system. We solve the problems by an indirect method, that is, we transform the quasianalytic system into an analytic system firstly, and the degenerate singular point into an elementary singular point. Then we calculate the singular values at the origin of the analytic system by the known classical methods. We obtain the center conditions and isochronous center conditions. Accordingly, integrability and pseudolinearizable conditions at degenerate singular point in the quasianalytic system are obtained. Especially, when λ = 1, the system has been studied in Wu and Zhang (2010).

The generalized center problem of resonant infinity for a polynomial differential system

This paper explores the problems of generalized center conditions and integrability of resonant infinity for a complex polynomial differential system. The method is based on converting resonant infinity into an elementary singular point by a homeomorphic transformation. The calculation of generalized singular point quantities is an effective way of finding necessary conditions for integrable systems. A new recursive algorithm for computing generalized singular point quantities at the origin of the transformed system is derived. At the same time, a necessary and sufficient condition for resonant infinity to be a generalized complex center is presented. As an application of the new recursive algorithm, the generalized center conditions for resonant infinity for a class of cubic systems are discussed. To the best of our knowledge, this is the first time that the generalized center problem has been considered for p : − q resonant infinity.

Bifurcation of Limit Cycles and Pseudo-Isochronicity at Degenerate Singular Point in a Septic System

This paper deals with the problems of bifurcation of limit cycles and pseudo-isochronous center conditions at degenerate singular point in a class of septic polynomial differential system. We solve the problems by an indirect method, i.e., we transform the degenerate singular point into an elementary singular point. Then we construct a septic system which allows the appearance of eight limit cycles in the neighborhood of degenerate singular point. Finally, we investigate the pseudo-isochronous center conditions at degenerate singular point for the system. As far as we know, this is the first time that an example of septic system with eight limit cycles bifurcating from degenerate singular point is given, and it is also the first time the pseudo-isochronous center conditions at degenerate singular point in a septic system are discussed.

Isochronicity Problem of Higher-Order Singular Point for Polynomial Differential Systems

Due to the difficulty, the isochronicity problems with respect to higher-order singular point (or degenerate singular point) of polynomial differential systems are far from being solved. The calculation of period constants is an effective way to find necessary conditions for isochronicity. In this paper, by means of a homeomorphic transformation, higher-order singular point is transferred into the origin. At the same time, a new recursive algorithm to compute period constants at the origin of the transformed system is deduced which is easy to realize with the computer algebraic system such as MATHEMATICA or MAPLE. Finally, to illustrate the effectiveness of our algorithm, the pseudo-isochronous center conditions of higher-order singular point for a class of septic system are investigated. Our work is new in terms of research about the isochronicity problem of higher-order singular point and consists of the existing results related to the origin as a special case when it is an elementary singular point.

Lie symmetries for the orbital linearization of smooth planar vector fields around singular points

This work is a generalization of the method proposed in [I.A. García, S. Maza, Linearization of analytic isochronous centers from a given commutator, J. Math. Anal. Appl. 339 (1) (2008) 740–745] of linearization of analytic isochronous centers from a given commutator. In this paper we propose a constructive procedure to get the change of variables that orbitally linearizes a smooth planar vector field on C 2 around an elementary singular point (i.e., a singular point with associated eigenvalues λ , μ ∈ C satisfying λ ≠ 0 ) or a nilpotent singular point from a given infinitesimal generator of a Lie symmetry.

Linearization of smooth planar vector fields around singular points via commuting flows

In this paper we propose a constructive procedure to get the change of variables that linearizes a smooth planar vector field on $\mathbb C^2$ around an elementary singular point (i.e., a singular point with associated eigenvalues $\lambda, \mu \in \mathbb C$ satisfying $\mu$≠$0$) or a nilpotent singular point from a given commutator. Moreover, it is proved that the near--identity change of variables that linearizes the vector field $\mathcal X = (x+\cdots) \partial_x + (y+\cdots) \partial_y$ is unique and linearizes simultaneously all the centralizers of $\mathcal X$. The method is used in order to obtain the linearization of some extracted examples of the existent literature.

The generalized center problem of degenerate resonant singular point

In this paper, generalized center condition and integrability of degenerate resonant singular point for a class of complex polynomial differential system were studied. The method was based on a homeomorphic transformation of the degenerate singular point into elementary singular point, which allows us to compute the generalized singular point quantities and determine the generalized center condition for the origin. In the end, we obtained the necessary and sufficient conditions of generalized complex center of degenerate resonant singular point.

Analytic classification of saddle nodes

Isolated degenerate elementary singular points (saddle nodes) of germs of holomorphic vector fields in $(\mathbb {C}^2,0)$ are studied. An analytic classification of them is obtained; it is shown that the analytic classification has two times more moduli (numeric and functional) than the orbital analytic classification. A theorem on sectorial normalization is proved.

Planar analytic vector fields with generalized rational first integrals

The main purpose of this paper is to characterize a germ of planar holomorphic vector field at an elementary singular point having a generalized rational first integral. Our results generalize a result due to Poincaré on a necessary condition of the existence of a rational first integral for planar polynomial systems. As two applications of our main result, we give the necessary and sufficient conditions on the existence of rational first integral for planar quadratic systems having either a weak nondegenerate singular point, or a degenerate elementary singular point.

Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics

In this paper we study the finite cyclicity of several elementary graphics appearing in quadratic systems. This makes substantial progress in the study of the finite cyclicity of the elementary graphics with non-identical return map listed in Dumortier et al J. Diff. Eqns 110 86 - 133 . The main tool we use is the method of Khovanskii. We also use the fact that some graphics have unbroken connections and we calculate normal forms for elementary singular points in the graphics. Several arguments use the fact that two singular points `compensate' each other precisely when the graphic surrounds a centre. One originality of the paper is to prove that for certain graphics among quadratic systems some regular transition maps are not tangent to the identity.

Smooth normal forms of folded elementary singular points

In this work the smooth and topological normal forms of the first-order implicit differential equations in the plane near its folded degenerate elementary singular point are found, and thereby, the smooth and topological classification of folded elementary singular points of these equations is completed. It is proved, for example, that the equation is equivalent near its folded singular point of saddlenode type to some equation (dy/dx+x 2+Ax 3)2=y, whereA is a real number, in some appropriate smooth coordinate system in the plane with the origin at this point. The numberA is the parameter of the normal form.

Orbital topological classification of analytic differential equations in a neighborhood of a degenerate elementary singular point on the two-dimensional complex plane

We consider the germ of an analytic differential equation in (C2, 0) defined by a vector field with a singularity at zero. A singular point of the equation is called degenerate elementary if one eigenvalue of the linear part of the field is equal to zero and the other is different from zero. The orbital topological classification of differential equations in a neighborhood of a degenerate elementary singular point is obtained for all cases except the so-called “Liouville” case. This case defines a subset of equations which does not separate the corresponding function space.

Condition for generating limit cycles by bifurcation of the loop of a saddle-point separatrix

In paper [1] (On the stability of a saddle-point separatrix loop and analytical criterion for its bifurcation limit cycles Acta Mathematica Sinica Vol. 28. No. 1, 55–70, Bejing China 1985), we considered the problem of generating limit cycles by the bifurcation of a stable or an unstable loop of a saddle-point separatrix. We gave for the first time a criterion for the stability of the loop as following:L0 is stable (unstable) if\(\int_{ - \infty }^\infty {(P'_{0x} + Q'_{0y} )dt 0)} \) wherex=ϕ(t),y=ϕ(t) then a sufficient condition for the bifurcation which generates limit cycles. This paper generalizes the result of [1] to the case where the loop contains a center or the loop tends to an infinite saddle-point, and removes the restriction that the saddle-point should be an elementary singular point. Applying the results of this paper, the author studies a two-parameter system $$\left\{ \begin{gathered} x = lx^2 + y^2 - y + 5\varepsilon xy \hfill \\ y = (3l + 5)xy + x + \varepsilon x^2 \hfill \\ \end{gathered} \right.$$